Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams

O. Thomas; A. Sénéchal; J.-F. Deü

doi:10.1007/s11071-016-2965-0

Nonlinear Dynamics

October 2016, Volume 86, Issue 2, pp 1293–1318 | Cite as

Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams

Authors
Authors and affiliations

O. Thomas
A. Sénéchal
J.-F. Deü

Original Paper

First Online: 24 August 2016

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12 Citations

Abstract

This work addresses the large amplitude nonlinear vibratory behavior of a rotating cantilever beam, with applications to turbomachinery and turbopropeller blades. The aim of this work is twofold. Firstly, we investigate the effect of rotation speed on the beam nonlinear vibrations and especially on the hardening/softening behavior of its resonances and the appearance of jump phenomena at large amplitude. Secondly, we compare three models to simulate the vibrations. The first two are based on analytical models of the beam, one of them being original. Those two models are discretized on appropriate mode basis and solve by a numerical following path method. The last one is based on a finite-element discretization and integrated in time. The accuracy and the validity range of each model are exhibited and analyzed.

Keywords

Large rotation Nonlinear von Karman Biot strain Inextensible Continuation method Asymptotic numerical method Modal expansion

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Notes

Acknowledgments

The French company Safran Snecma and the French Ministry of Research are thanked for the financial support of this study, through the PhD grant of the second author. Émmanuel Cottanceau is also warmly thanked for the careful reading of the finite-element model details section.

Appendix 1: Nonlinear coefficients for the inextensible model

The coefficients $A_p^k$, $B_{p}^k$, $\Gamma _{pqr}^k$, $\Lambda _{pqr}^k$ et $\Pi _{pqr}^k$ and the modal forcing $p_k$ are given by

$$\begin{aligned}&A_{p}^{k}= \int _{0}^{1}{\left[ \left( \Phi '_{p}e_{s}\right) ''+\Phi '''_{p}e_{s}\right] '\Phi _{k}}\,{\text {d}}X,\end{aligned}$$

(72a)

$$\begin{aligned}&B_{p}^{k}=\int _{0}^{1}\left[ \left( \Phi '_{p}\int _{1}^{X}{\int _{0}^{X}{e_{s}}\,\text {d}X}{\text {d}}X\right) '\right. \nonumber \\&\quad \left. -\left( \Phi '_{p}\left( 1- e_{s}\right) \int _{1}^{X}{\left( R+X\right) }\,{\text {d}}X\right) '\right] \Phi _{k}\,{\text {d}}X ,\end{aligned}$$

(72b)

$$\begin{aligned}&\Gamma _{pqr}^{k}=\int _{0}^{1}{\left[ \Phi '_{p}\Phi ''_{q}\Phi ''_{r}+\Phi '''_{p}\Phi '_{q}\Phi '_{r}\right] '\Phi _{k}}\,{\text {d}}X ,\end{aligned}$$

(72c)

$$\begin{aligned}&\Lambda _{pqr}^{k}=\int _{0}^{1}\left[ \Phi '_{p}\Phi '_{q}\Phi '_{r}\int _{1}^{X}{\left( R+X\right) }\,{\text {d}}X\right. \nonumber \\&\left. -\Phi '_{p}\int _{1}^{X}\int _{0}^{X}{\Phi '_{q}\Phi '_{r}}\,{\text {d}}X{\text {d}}X \right] '\Phi _{k}\,{\text {d}}X, \end{aligned}$$

(72d)

$$\begin{aligned}&\Pi _{pqr}^{k}=\int _{0}^{1}{\left[ \Phi '_{p}\int _{1}^{X}{\int _{0}^{X}{\Phi '_{q}\Phi '_{r}}}\,{\text {d}}X{\text {d}}X\right] '\Phi _{k}}\,{\text {d}}X, \end{aligned}$$

(72e)

$$\begin{aligned}&p_k=\int _0^1 p\,\Phi _k {\text {d}}X. \end{aligned}$$

(72f)

Appendix 2: Finite-element details

Details of the beam finite-element discretization of Sect. 4 are specified here, by starting from the variational formulation of Eq. (22). In the following, a superscript or a subscript e refers to elementary quantities. The generalized displacements are discretized using linear shape functions. Thus, axial displacement, transverse displacements and fiber rotations on one finite element of length $L_e$ are gathered into $\mathbf {u}^e=[u\;w\;\theta ]^{{\text {T}}}$ and related to the elementary dofs vector $\mathbf {q}^e$ by, for all $X\in [0 \;L_e]$:

$$\begin{aligned} \mathbf {u}^e(X) = \mathbf {N}(X)\,\mathbf {q}^e \end{aligned}$$

(73)

where $\mathbf {q}^e=[u_1\;w_1\;\theta _1\;u_2\;w_2\;\theta _2]^{{\text {T}}}$,

$$\begin{aligned} \mathbf {N}(X)=\left[ \begin{array}{cccccc} N_{1} &{} 0 &{} 0 &{} N_{2} &{} 0 &{} 0 \\ 0 &{} N_{1} &{} 0 &{} 0 &{} N_{2} &{} 0 \\ 0 &{} 0 &{} N_{1} &{} 0 &{} 0 &{} N_{2} \end{array} \right] , \end{aligned}$$

(74)

and $u_1$, $w_1$ and $\theta _1$ (respectively, $u_2$, $w_2$ and $\theta _2$) correspond to the first (respectively, second) node of the beam element. The two shape functions are:

$$\begin{aligned} N_1(X)=1-\frac{X}{L_e},\quad N_2(X)=\frac{X}{L_e}. \end{aligned}$$

(75)

Therefore, the discretized expressions of strains e, $\gamma $ and $\kappa $ (Eqs. (15a–c)) are found to be:

$$\begin{aligned} e^e&=\left( 1+\frac{u_2-u_1}{L}\right) \cos \theta + \left( \frac{w_2-w_1}{L}\right) \sin \theta -1, \end{aligned}$$

(76a)

$$\begin{aligned} \gamma ^e&= \left( \frac{w_2-w_1}{L}\right) \cos \theta - \left( 1+\frac{u_2-u_1}{L}\right) \sin \theta ,\end{aligned}$$

(76b)

$$\begin{aligned} \kappa ^e&= \frac{\theta _2-\theta _1}{L}. \end{aligned}$$

(76c)

They are gathered into the elementary strain vector $\mathbf {e}=[e^e\;\gamma ^e\;\kappa ^e]^{{\text {T}}}$ which can be written as $\mathbf {e}(X)=\mathbf {B}(X,\mathbf {q}^e)\mathbf {q}^e$, with $\mathbf {B}$ the elementary discretized gradient matrix.

The works of the internal forces, the external forces and the acceleration forces [Eqs. (23a–c)] are discretized as follows:

$$\begin{aligned} \delta \mathcal {W}_a&= \int _0^{L_e} \delta \mathbf {u}^{e{\text {T}}}\,\mathbf {J}\,\ddot{\mathbf {u}}^e = \delta \mathbf {q}^{e{\text {T}}} \underbrace{\left( \int _0^{L_e}\mathbf {N}^{{\text {T}}}\mathbf {J}\mathbf {N}\,\text {d}X\right) }_{\mathbf {M}^e}\mathbf {q}^e, \\ \delta \mathcal {W}_i&= - \int _0^{L_e} \left( EAe\delta e + kG\gamma \delta \gamma + EI\kappa \delta \kappa \right) \,\text {d}X \\&= - \int _0^{L_e}\delta \mathbf {e}^{{\text {T}}}\mathbf {C}\mathbf {e}\,\text {d}X \\&=- \delta \mathbf {q}^{e{\text {T}}} \underbrace{\left( \int _0^{L_e}\mathbf {B}^{{\text {T}}}\mathbf {C}\mathbf {B}\mathbf {q}^e\,\text {d}X\right) }_{\mathbf {f}_\text {int}^e(\mathbf {q}^e)},\\ \delta \mathcal {W}_e&=\underbrace{\rho \Omega ^2 \int _0^{L_e}\left[ A\left( R+X+u\right) \;0 \; -\frac{1}{2}I\sin \left( 2\theta \right) \right] \mathbf {N}\,\text {d}X}_{\mathbf {f}_\Omega ^{e{\text {T}}}}\,\delta \mathbf {q}^e \\&+ \underbrace{\int _0^{L_e} \left[ n \;\; p \;\; q\right] \mathbf {N}\,\text {d}X}_{\mathbf {f}_\text {ext}^{{\text {T}}}}\,\delta \mathbf {q}^e. \end{aligned}$$

where

$$\begin{aligned} \mathbf {J}=\left[ \begin{array}{ccc} A &{} 0 &{} 0 \\ 0 &{} A &{} 0 \\ 0 &{} 0 &{} I \\ \end{array} \right] \quad \text {and}\quad \mathbf {C}=\left[ \begin{array}{ccc} EA &{} 0 &{} 0 \\ 0 &{} kGA &{} 0 \\ 0 &{} 0 &{} EI \\ \end{array} \right] . \end{aligned}$$

The elementary mass matrix is obtained by directly evaluating the above integral, to obtain:

$$\begin{aligned} \mathbf {M}^e=\frac{\rho L_e}{6} \left[ \begin{array}{cccccc} 2A &{} 0 &{} 0 &{} A &{} 0 &{} 0 \\ 0 &{} 2A &{} 0 &{} 0 &{} A &{} 0 \\ 0 &{} 0 &{} 2I &{} 0 &{} 0 &{} I \\ A &{} 0 &{} 0 &{} 2A &{} 0 &{} 0 \\ 0 &{} A &{} 0 &{} 0 &{} 2A &{} 0 \\ 0 &{} 0 &{} I &{} 0 &{} 0 &{} 2I \end{array} \right] . \end{aligned}$$

(77)

The integral for the internal force vector is evaluated by thanks to a reduced integration with the one-point Gauss rule at $X = L_e/2$ to avoid shear locking [6, Sect. 5.4.1]. We denote by $\bar{\theta }=(\theta _2+\theta _1)/2$, $\bar{c}=\cos \bar{\theta }$ and $\bar{s}=\sin \bar{\theta }$ the values of $\cos \theta $ and $\sin \theta $ at $X = L_e/2$. One thus obtains:

$$\begin{aligned} \mathbf {f}_\text {int}^e= & {} EA\bar{e} \left[ \begin{array}{c} -\bar{c} \\ -\bar{s} \\ \bar{\gamma }L_e/2\\ \bar{c}\\ \bar{s}\\ \bar{\gamma }L_e/2 \end{array} \right] +kGA\bar{\gamma } \left[ \begin{array}{c} \bar{s} \\ -\bar{c} \\ -\left( 1+\bar{e}\right) L_e/2\\ -\bar{s}\\ \bar{c}\\ -\left( 1+\bar{e}\right) L_e/2 \end{array} \right] \nonumber \\&+EI\bar{\kappa } \left[ \begin{array}{c} 0\\ 0 \\ -1\\ 0\\ 0\\ 1 \end{array} \right] , \end{aligned}$$

(78)

where $\bar{e}=e^e(\bar{\theta })$, $\bar{\gamma }=\gamma ^e(\bar{\theta })$ and $\bar{\kappa }=\kappa ^e(\bar{\theta })$. Then, the integral in the centrifugal force vector $\mathbf {f}_\Omega $ is evaluated with the average angle $\bar{\theta }$. The reason is that $ \theta = N_{1}\theta _{1} + N_{2}\theta _{2}$, which is the exact expression of $\theta $ coming from the finite-element discretization, gives efforts as a fraction whose denominator is $\left( \theta _{1} - \theta _{2} \right) ^2$. Therefore, it may lead to tremendous efforts when the beam passes through its initial position $\left( \theta _{1} \approx \theta _{2}\right) $, whereas it is not the case since the initial position is the one where the only effort involved is those due to rotation. Using this reduced integration only for $\theta $ leads to obtain:

$$\begin{aligned} \mathbf {f}_\Omega ^e= & {} \underbrace{ \frac{1}{4}\rho I\Omega ^{2}L_e \left[ \begin{array}{c} 0 \\ 0 \\ \sin \left( 2\bar{\theta }\right) \\ 0 \\ 0 \\ \sin \left( 2\bar{\theta }\right) \end{array} \right] }_{\mathbf {f}_{\Omega }^{\text {nl}}} + \underbrace{ \frac{1}{6}\rho A\Omega ^{2}L_e \left[ \begin{array}{c} 2u_{1}+u_{2} \\ 0 \\ 0 \\ u_{1}+2u_{2} \\ 0 \\ 0 \end{array} \right] }_{\mathbf {f}_{\Omega }^{\text {lin}}}\nonumber \\&+ \underbrace{ \frac{1}{6}\rho A\Omega ^{2}L_e \left[ \begin{array}{c} L+3R \\ 0 \\ 0 \\ 2L+3R \\ 0 \\ 0 \end{array} \right] }_{\mathbf {f}_{\Omega }^{\text {cste}}} \end{aligned}$$

(79)

The elementary stiffness matrix, used for the Newton–Raphson algorithm iterations, is obtained in the following way. It is obtained by differentiating the internal force vector and the centrifugal force vector:

$$\begin{aligned} \mathbf {K}_\text {t}=\frac{\partial \mathbf {f}}{\partial \mathbf {q}}=\frac{\partial \mathbf {f}_\text {int}}{\partial \mathbf {q}}-\frac{\partial \mathbf {f}_\Omega }{\partial \mathbf {q}}=\mathbf {K}_\text {mat} + \mathbf {K}_\text {geo} - \mathbf {K}_\text {c} - \mathbf {K}_\text {gc}.\nonumber \\ \end{aligned}$$

(80)

This matrix is the sum of the material stiffness $\mathbf {K}_\text {mat}$, the geometric stiffness $\mathbf {K}_\text {geo}$, the centrifugal stiffness $\mathbf {K}_\text {c}$ and the geometric centrifugal stiffness $\mathbf {K}_\text {gc}$. Among these four contributions to the total stiffness, $\mathbf {K}_\text {geo}$ and $\mathbf {K}_\text {gc}$ are deflection dependent, whereas $\mathbf {K}_\text {mat}$ and $\mathbf {K}_\text {c}$ are linear. The material stiffness comes from the variation of the nodal displacement $\delta \mathbf {q}$ of the generalized forces while keeping the discretized gradient $\mathbf {B}$ fixed and the geometric stiffness comes from the variation of $\mathbf {B}$ while the generalized forces are kept fixed. The centrifugal stiffness stands for the constant additional stiffness due to the rotation, and the geometric centrifugal stiffness represents the deflection-dependent centrifugal effect that depends on the rotation angle $\theta $.

On then obtains, at the elementary level:

$$\begin{aligned} \frac{\partial \mathbf {f}_\text {int}^e}{\partial \mathbf {q}^e}= & {} \underbrace{\int _0^{L_e}\mathbf {B}^{{\text {T}}}\mathbf {C}\mathbf {B}\,\text {d}X}_{\mathbf {K}_{\text {mat}}} + \underbrace{\int _0^{L_e}\frac{\partial \mathbf {B}^{{\text {T}}}}{\partial \mathbf {q}^e}\mathbf {C}\mathbf {B}\mathbf {q}^e\,\text {d}X}_{\mathbf {K}_{\text {geo}}}\nonumber \\= & {} \mathbf {K}_e^e+\mathbf {K}_\gamma ^e+ \mathbf {K}_\kappa ^e \end{aligned}$$

(81)

These three contributions correspond to the tangent stiffness matrix due to the axial strain e, the shear $\gamma $ and the curvature $\kappa $, which write:

$$\begin{aligned} \mathbf {K}_e^{e}= & {} \frac{EA}{L} \left[ \begin{array}{rrrrrr} K_{1} &{} K_{3} &{} K_{4} &{} -K_{1} &{} -K_{3} &{} K_{4} \\ &{} K_{2} &{} K_{5} &{} -K_{3} &{} -K_{2} &{} K_{5} \\ &{} &{} K_{6} &{} -K_{4} &{} -K_{5} &{} K_{6} \\ &{} &{} &{} K_{1} &{} K_{3} &{} -K_{4} \\ &{} &{} &{} &{} K_{2} &{} -K_{5} \\ &{} &{} &{} &{} &{} K_{6} \end{array} \right] , \\ \mathbf {K}_\gamma ^e= & {} \frac{kGA}{L} \left[ \begin{array}{rrrrrr} K_{2} &{} -K_{3} &{} K_{7} &{} -K_{2} &{} K_{3} &{} K_{7} \\ &{} K_{1} &{} K_{8} &{} K_{3} &{} -K_{1} &{} K_{8} \\ &{} &{} K_{9} &{} -K_{7} &{} -K_{8} &{} K_{9} \\ &{} &{} &{} K_{2} &{} -K_{3} &{} -K_{7} \\ &{} &{} &{} &{} K_{1} &{} -K_{8} \\ &{} &{} &{} &{} &{} K_{9} \end{array} \right] ,\\ \quad \mathbf {K}_\kappa ^e= & {} \frac{EI}{L} \left[ \begin{array}{rrrrrr} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ &{} &{} 1 &{} 0 &{} 0 &{}-1 \\ &{} &{} &{} 0 &{} 0 &{} 0 \\ &{} &{} &{} &{} 0 &{} 0 \\ &{} &{} &{} &{} &{} 1 \end{array} \right] , \end{aligned}$$

with

$$\begin{aligned} K_{1}= & {} \bar{c}^2, \quad K_{2}=\bar{s}^2, \quad K_{3}=\bar{c}\bar{s},\\ \quad K_{4}= & {} \frac{L_e}{2}\left( \bar{e}\bar{s}-\bar{\gamma }\bar{c}\right) , \quad K_{5}=-\frac{L_e}{2}\left( \bar{e}\bar{c}+\bar{\gamma }\bar{s}\right) , \\ K_{6}= & {} \frac{L_e^{2}}{4}\left[ \bar{\gamma }^2-\bar{e}\left( \bar{e}+1\right) \right] ,\\ \quad K_{7}= & {} \frac{L_e}{2}\left[ \bar{\gamma }\bar{c}-\left( \bar{e}+1\right) \bar{s}\right] , \\ K_{8}= & {} \frac{L_e}{2}\left[ \bar{\gamma }\bar{s}+\left( \bar{e}+1\right) \bar{c}\right] , \quad \\ K_{9}= & {} \frac{L_e^{2}}{4}\left[ \left( \bar{e}+1\right) ^2-\bar{\gamma }^2\right] . \end{aligned}$$

The centrifugal stiffness $\mathbf {K}^e_\text {c}$ and the geometric centrifugal stiffness $\mathbf {K}^e_\text {gc}$ come from

$$\begin{aligned} \frac{\partial \mathbf {f}^e_\Omega }{\partial \mathbf {q}^e}=\mathbf {K}^e_\text {c}+\mathbf {K}^e_\text {gc}, \end{aligned}$$

(83)

with

$$\begin{aligned} \mathbf {K}^e_\text {c}&=\frac{1}{6}\rho AL\Omega ^2 \left[ \begin{array}{cccccc} 2 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ &{} &{} 0 &{} 0 &{} 0 &{} 0 \\ &{} &{} &{} 2 &{} 0 &{} 0 \\ &{} &{} &{} &{} 0 &{} 0 \\ &{} &{} &{} &{} &{} 0 \end{array} \right] , \\ \quad \mathbf {K}^e_\text {gc}&=\frac{1}{4}\rho IL\Omega ^2\cos 2\bar{\theta } \left[ \begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ &{} &{} 1 &{} 0 &{} 0 &{} 1 \\ &{} &{} &{} 0 &{} 0 &{} 0 \\ &{} &{} &{} &{} 0 &{} 0 \\ &{} &{} &{} &{} &{} 1 \end{array} \right] . \end{aligned}$$

(84a)

References

1.
Antman, S.S., Kenney, C.S.: Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity. Arch. Ration. Mech. Anal. 76(4), 289–338 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
2.
Apiwattanalunggarn, P., Shaw, S.W., Pierre, C., Jiang, D.: Finite-element-based nonlinear modal reduction of a rotating beam with large-amplitude motion. J. Vib. Control 9(3–4), 235–263 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
3.
Arquier, R., Karkar, S., Lazarus, A., Thomas, O., Vergez, C., Cochelin, B. : Manlab 2.0: an interactive path-following and bifurcation analysis software. Technical report, Laboratoire de Mécanique et d’Acoustique, CNRS, http://manlab.lma.cnrs-mrs.fr, (2005-2011)
4.
Arvin, H., Bakhtiari-Nejad, F.: Non-linear modal analysis of a rotating beam. Int. J. Non-Linear Mech. 46(6), 877–897 (2011)CrossRefGoogle Scholar
5.
Austin, F., Pan, H.H.: Planar dynamics of free rotating flexible beams with tip masses. Am. Inst. Aeronaut. Astronaut. J. 8, 726–733 (1970)CrossRefzbMATHGoogle Scholar
6.
Bathe, K.-J.: Finite Element Procedures. Prentice Hall, Upper Saddle River (1996)zbMATHGoogle Scholar
7.
Bauchau, O., Guernsey, D.: On the choice of appropriate bases for nonlinear dynamic modal analysis. J. Am. Helicopter Soc. 38(4), 28–36 (1993)CrossRefGoogle Scholar
8.
Bauchau, O.A., Hong, C.H.: Finite element approach to rotor blade modeling. J. Am. Helicopter Soc. 32(1), 60–67 (1987)CrossRefGoogle Scholar
9.
Bauchau, O.A., Hong, C.H.: Nonlinear response and stability analysis of beams using finite elements in time. AIAA J. 26(9), 1135–1142 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
10.
Bauer, H.F., Eidel, W.: Vibration of a rotating uniform beam, part ii: Orientation perpendicular to the axis of rotation. J. Sound Vib. 122, 357–375 (1988)CrossRefGoogle Scholar
11.
Bazoune, A.: Survey on modal frequencies of centrifugally stiffened beams. Shock Vib. Dig. 37, 449–469 (2005)CrossRefGoogle Scholar
12.
Bazoune, A., Khulief, Y.A.: Furthur results for modal characteristics of rotating tapered timoshenko beams. J. Sound Vib. 219(1), 157–174 (1999)CrossRefGoogle Scholar
13.
Bekhoucha, F., Rechak, S., Duigou, L., Cadou, J.-M.: Nonlinear forced vibrations of rotating anisotropic beams. Nonlinear Dyn. 74(4), 1281–1296 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
14.
Boyce, W.E.: Effect of hub radius on the vibrations of a uniform bar. J. Appl. Mech. 23, 287–290 (1956)zbMATHGoogle Scholar
15.
Cao, D.Q., Tucker, R.W.: Nonlinear dynamics of elastic rods using the cosserat theory: modelling and simulation. Int. J. Solids Struct. 45(2), 460–477 (2008)CrossRefzbMATHGoogle Scholar
16.
Cochelin, B., Vergez, C.: A high order purely frequential harmonic balance formulation. J. Sound Vib. 324(1–2), 243–262 (2009)CrossRefGoogle Scholar
17.
Crespo da Silva, M.R.M., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams. I. equations of motion. J. Struct. Mech. 6, 437–448 (1978)CrossRefGoogle Scholar
18.
Crespo da Silva, M.R.M., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams. II. forced motions. J. Struct. Mech. 6, 449–461 (1978)CrossRefGoogle Scholar
19.
Crespo Da Silva, M.R.M., Hodges, D.H.: Nonlinear flexure and torsion of rotating beams, with application to helicopter rotor blades—I. Formulation. Vertica 10(2), 151–169 (1986)Google Scholar
20.
Cusumano, J.P., Moon, F.C.: Chaotic non-planar vibrations of the thin elastica, part 2: derivation and analysis of a low-dimensional model. J. Sound Vib. 179(2), 209–226 (1995)CrossRefGoogle Scholar
21.
Danielson, D.A., Hodges, D.H.: Nonlinear beam kinematics by decomposition of the rotation tensor. J. Appl. Mech. 54(2), 258–262 (1987)CrossRefzbMATHGoogle Scholar
22.
Das, S.K., Ray, P.C., Pohit, G.: Free vibration of a rotating beam with nonlinear spring and mass system. J. Sound Vib. 301, 165–188 (2007)CrossRefzbMATHGoogle Scholar
23.
Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci. 44(1), 1–23 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
24.
Eringen, A.C.: On the non-linear vibration of elastic bars. Q. Appl. Math. 9, 361–369 (1952)MathSciNetzbMATHGoogle Scholar
25.
Felippa, C.: Nonlinear Finite Element Methods, chapter 10: The TL Plane Beam Element: formulation. http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d, (2012)
26.
Géradin, M., Cardona, A.: Flexible Multibody Dynamics: a Finite Element Approach. Wiley, New York (2001)Google Scholar
27.
Géradin, M., Rixen, D.: Mechanical Vibrations. Theory and Application to Structural Dynamics. Wiley, New York (1997)Google Scholar
28.
Gerstmayr, J., Irschik, H.: On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach. J. Sound Vib. 318(3), 461–487 (2008)CrossRefGoogle Scholar
29.
Hamdan, M.N., Al-Bedoor, B.O.: Non-linear free vibration of a rotating flexible arm. J. Sound Vib. 242(5), 839–853 (2001)CrossRefzbMATHGoogle Scholar
30.
Hodges, D.H.: Nonlinear beam kinematics for small strains and finite rotations. Vertica 11(3), 573–589 (1987)Google Scholar
31.
Hodges, D.H.: Geometrically-exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA J. 41(6), 1131–1137 (2003)CrossRefGoogle Scholar
32.
Hodges, D.H.: Nonlinear Composite Beam Theory. American Institute of Aeronautics and Astronautics, Reston (2006)CrossRefGoogle Scholar
33.
Hsieh, S.-R., Shaw, S.W., Pierre, C.: Normal modes for large amplitude vibration of a cantilever beam. Int. J. Solids Struct. 31(14), 1981–2014 (1994)CrossRefzbMATHGoogle Scholar
34.
Hutchinson, J.R.: Shear coefficients for timoshenko beam theory. J. Appl. Mech. 68(1), 87–92 (2001)CrossRefzbMATHGoogle Scholar
35.
Irschik, H., Gerstmayr, J.: A continuum mechanics based derivation of reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight bernoulli-euler beams. Acta Mech. 206(1–2), 1–21 (2009)CrossRefzbMATHGoogle Scholar
36.
Irschik, H., Gerstmayr, J.: A continuum-mechanics interpretation of reissner’s non-linear shear-deformable beam theory. Math. Comput. Model. Dyn. Syst. 17(1), 19–29 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
37.
Jiang, D., Pierre, C., Shaw, S.W.: The construction of non-linear normal modes for systems with internal resonance. Int. J. Non-Linear Mech. 40(5), 729–746 (2005)CrossRefzbMATHGoogle Scholar
38.
Jones, L.H.: The transverse vibration of a rotating beam with tip mass: the method of integral equations. Q. Appl. Math. 33, 193–203 (1975)zbMATHGoogle Scholar
39.
Lacarbonara, W., Arvin, H., Bakhtiari-Nejad, F.: A geometrically exact approach to the overall dynamics of elastic rotating blades - part 2: flapping nonlinear normal modes. Nonlinear Dyn. 70(3), 2279–2301 (2012)MathSciNetCrossRefGoogle Scholar
40.
Lacarbonara, W., Yabuno, H.: Refined models of elastic beams undergoing large in-plane motions: theory and experiment. Int. J. Solids Struct. 43, 5066–5084 (2006)CrossRefzbMATHGoogle Scholar
41.
Lazarus, A., Miller, J.T., Reis, P.M.: Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method. J. Mech. Phys. Solids 61(8), 1712–1736 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
42.
Lazarus, A., Thomas, O., Deü, J.-F.: Finite elements reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS. Finite Elem. Anal. Des. 49(1), 35–51 (2012)MathSciNetCrossRefGoogle Scholar
43.
Lo, H., Goldberg, J.E., Bogdanoff, J.L.: Effect of small hub radius change on bending frequencies of a rotating beam. J. Appl. Mech. 27, 548–550 (1960)CrossRefzbMATHGoogle Scholar
44.
Lo, H., Renbarger, J.L.: Bending Vibrations of a Rotating Beam. In First US National Congress of Applied Mechanics, Chicago, Illinois (1951)Google Scholar
45.
Magnusson, A., Ristinmaa, M., Ljun, C.: Behaviour of the extensible elastica solution. Int. J. Solids Struct. 38(46–47), 8441–8457 (2001)CrossRefzbMATHGoogle Scholar
46.
Marguerre, K. : Zur Theorie der Gekrümmten Platte Grosser Formänderung. In: Proceedings of the 5th International Congress for Applied Mechanics, pp. 93–101, (1938)Google Scholar
47.
Mettler, E.: Zum problem der stabilität erzwungener schwingungen elastischer körper. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 31(8–9), 263–264 (1951)CrossRefzbMATHGoogle Scholar
48.
Nayfeh, A.H.: Nonlinear transverse vibrations of beams with properties that vary along the length. J. Acoust. Soc. Am. 53(3), 766–770 (1973)CrossRefGoogle Scholar
49.
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)zbMATHGoogle Scholar
50.
Nayfeh, A.H., Paï, P.F.: Linear and Nonlinear Structural Mechanics. Wiley, New York (2004)CrossRefzbMATHGoogle Scholar
51.
Nayfeh, A.H., Pai, P.F.: Non-linear non-planar parametric responses of an inextensional beam. Int. J. Non-Linear Mech. 24(2), 139–158 (1989)CrossRefzbMATHGoogle Scholar
52.
Noijen, S.P.M., Mallon, N.J., Fey, R.H.B., Nijmeijer, H., Zhang, G.Q.: Periodic excitation of a buckled beam using a higher order semianalytic approach. Nonlinear Dyn. 50(1–2), 325–339 (2007)CrossRefzbMATHGoogle Scholar
53.
Ogden, R.W.: Non-linear Elastic Deformations. Dover, New York (1997)Google Scholar
54.
Pai, P.F., Nayfeh, A.H.: Non-linear non-planar oscillations of a cantilever beam under lateral base excitations. Int. J. Non-Linear Mech. 25(5), 455–474 (1990)MathSciNetCrossRefGoogle Scholar
55.
Pai, P.F., Palazotto, A.N.: Large-deformation analysis of flexible beams. Int. J. Solids Struct. 33(9), 1335–1353 (1996)CrossRefzbMATHGoogle Scholar
56.
Palacios, R.: Nonlinear normal modes in an intrinsic theory of anisotropic beams. J. Sound Vib. 330(8), 1772–1792 (2011)CrossRefGoogle Scholar
57.
Park, J.-H., Kim, J.-H.: Dynamic analysis of rotating curved beam with tip mass. Journal Sound Vib. 228, 1017–1034 (1999)CrossRefGoogle Scholar
58.
Pesheck, E., Pierre, C., Shaw, S.W.: Modal reduction of a nonlinear rotating beam through nonlinear normal modes. J. Vib. Acoust. 124, 229–236 (2002)CrossRefGoogle Scholar
59.
Reissner, E.: On one-dimensional finite strain beam theory: the plane problem. Z. Angew. Math. Phys. 23(5), 795–804 (1972)CrossRefzbMATHGoogle Scholar
60.
Rosen, A.: Structural and dynamic behaviour of pre-twisted rods and beams. Appl. Mech. Rev. 44, 483–515 (1991)CrossRefGoogle Scholar
61.
Sansour, C., Sansour, J., Wriggers, P.: A finite element approach to the chaotic motion of geometrically exact rods undergoing in-plane deformations. Nonlinear Dyn. 11(2), 189–212 (1996)MathSciNetCrossRefGoogle Scholar
62.
Simo, J.C., Vu-Quoc, L.: On the dynamics of rods undergoing large motions - a geometrically exact approach. Comput. Methods Appl. Mech. Eng. 66, 125–161 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
63.
Sinha, S.K.: Combined torsional-bending-axial dynamics of a twisted rotating cantilever timoshenko beam with contact-impact loads at the free end. J. Appl. Mech. 74, 505–522 (2007)CrossRefzbMATHGoogle Scholar
64.
Sokolov, I., Krylov, S., Harari, I.: Electromechanical analysis of micro-beams based on planar finite-deformation theory. Finite Elem. Anal. Des. 49(1), 28–34 (2012)MathSciNetCrossRefGoogle Scholar
65.
Stoykov, S., Ribeiro, P.: Vibration analysis of rotating 3d beams by the p-version finite element method. Finite Elem. Anal. Des. 65, 76–88 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
66.
Tabaddor, M.: Influence of nonlinear boundary conditions on the single-mode response of a cantilever beam. Int. J. Solids Struct. 37, 4915–4931 (2000)CrossRefzbMATHGoogle Scholar
67.
Thomas, O., Bilbao, S.: Geometrically non-linear flexural vibrations of plates: in-plane boundary conditions and some symmetry properties. J. Sound Vib. 315(3), 569–590 (2008)CrossRefGoogle Scholar
68.
Thomas, O., Lazarus, A., Touzé, C.: A harmonic-based method for computing the stability of periodic oscillations of non-linear structural systems. In Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2010, Montreal, Canada, August 2010Google Scholar
69.
Thomas, O., Touzé, C., Chaigne, A.: Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance. Int. J. Solids Struct. 42(11–12), 3339–3373 (2005)CrossRefzbMATHGoogle Scholar
70.
Trindade, M.A., Sampaio, R.: Dynamics of beams undergoing large rotations accounting for arbitrary axial deformation. J. Guid. Control Dyn. 25(4), 634–643 (2002)CrossRefGoogle Scholar
71.
Turhan, Ö., Bulut, G.: On nonlinear vibrations of a rotating beam. J. Sound Vib. 322, 314–335 (2009)CrossRefGoogle Scholar
72.
von Karman, Th: Festigkeitsprobleme im maschinenbau. Encyklop adie der Mathematischen Wissenschaften 4(4), 311–385 (1910)zbMATHGoogle Scholar
73.
Whitman, A.B., DeSilva, C.N.: A dynamical theory of elastic directed curves. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 20(2), 200–212 (1969)CrossRefzbMATHGoogle Scholar
74.
Woinowsky-Krieger, S.: The effect of axial force on the vibration of hinged bars. J. Appl. Mech. 17, 35–36 (1950)MathSciNetzbMATHGoogle Scholar
75.
Woodall, S.R.: On the large amplitude oscillations of a thin elastic beam. Int. J. Non-linear Mech. 1, 217–238 (1966)CrossRefzbMATHGoogle Scholar
76.
Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin (2008)zbMATHGoogle Scholar
77.
Wright, A.D., Smith, C.E., Thresher, R.W., Wang, J.L.C.: Vibration modes of centrifugally stiffened beams. J. Appl. Mech. 49(1), 197–202 (1982)CrossRefzbMATHGoogle Scholar
78.
Wu, G., He, X., Pai, P.F.: Geometrically exact 3d beam element for arbitrary large rigid-elastic deformation analysis of aerospace structures. Finite Elem. Anal. Des. 47, 402–412 (2011)CrossRefGoogle Scholar

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O. Thomas
- 1
Email author
A. Sénéchal
- 2
- 3
J.-F. Deü
- 2

1.LSIS UMR CNRS 7296Arts et Métiers ParisTechLilleFrance
2.Structural Mechanics and Coupled Systems Laboratory, CnamParisFrance
3.Airbus Defence and SpaceLes MureauxFrance

Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams

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